PlanetPhysics/General Dynamic Systems

General dynamic system descriptions as stable space-time structures

Introduction: General system description

A general system can be described as a dynamical `whole', or entity capable of maintaining its working conditions; more precise system definitions are as follows.

A simple system is in general a bounded , but not necessarily closed, entity-- here represented as a [[../Cod/|category]] of stable, interacting components with inputs and outputs from the system's environment, or as a [[../SuperCategory6/|supercategory]] for a complex system consisting of subsystems, or components, with internal boundaries among such subsystems. In order to define a system one therefore needs to specify the following data:

1. components or subsystems;
2. mutual interactions, [[../Bijective/|relations]] or links;
3. a separation of the selected system by some boundary which distinguishes the system from its environment, without necessarily `closing' the system to material exchange with its environment;
4. the specification of the system's environment;
5. the specification of the system's categorical structure and [[../MathematicalFoundationsOfQuantumTheories/|dynamics]] (a supercategory will be required only when either the components or subsystems need be themselves considered as represented by a category , i.e. the system is in fact a super-system of (sub)systems, as it is the case of emergent super-complex systems or organisms).

Remarks

Point (5) claims that a system should occupy either a macroscopic or a microscopic [[../SR/|space-time]] region, but a system that comes into birth and dies off extremely rapidly may be considered either a short-lived process, or rather, a `[[../QualityFactorOfAResonantCircuit/|resonance]]' --an instability rather than a system, although it may have significant effects as in the case of `virtual [[../Particle/|particles',]] `virtual photons', etc., as in [[../QED/|quantum electrodynamics]] and chromodynamics. Note also that there are many other, different mathematical definitions of systems, ranging from (systems of) coupled [[../DifferentialEquations/|differential equations]] to [[../QuantumSpinNetworkFunctor2/|operator]] formulations, [[../TrivialGroupoid/|semigroups]], [[../TrivialGroupoid/|monoids]], [[../GroupoidHomomorphism2/|topological groupoid]] dynamic systems and dynamic categories. Clearly, the more useful system definitions include [[../CoIntersections/|algebraic]] and/or [[../TrivialGroupoid/|topological structures]] rather than simple, discrete structure sets, classes or their categories. The main intuition behind this first understanding of system is well expressed by the following passage: The most general and fundamental property of a system is theinter-dependence of parts/components/sub-systems or variables.

The inter-dependence relation consists in the existence of a family of determinate relationships among the parts or variables as contrasted with randomness or extreme variability. In other words, inter-dependence is the presence or existence of a certain organizational order in the relationship among the components or subsystems which make up the system. It can be shown that such organizational order must either result in a stable attractor or else it should occupy a stable space-time [[../Bijective/|domain]], which is generally expressed in closed systems by the [[../PreciseIdea/|concept]] of [[../ThermalEquilibrium/|equilibrium]].

On the other hand, in non-equilibrium, [[../ThermodynamicLaws/|open systems]], such as living systems, one cannot have a [[../Statics/|static]] but only a dynamic self-maintenance in a `state-space region' of the open system -- which cannot degenerate to either an equilibrium state or a single attractor space-time region. Thus, non-equilibrium, open systems that are capable of self-maintenance will also be generic, or structurally-stable : their arbitrary, small perturbation from a homeostatic maintenance regime does not result either in completely chaotic dynamics with a single attractor or the loss of their stability. It may however involve an ordered process of change - a process that follows a determinate, multi-stable pattern rather than random variation relative to the starting point.

General dynamic system definition

A formal (but natural) definition of a general dynamic system , either simple or complex can also be specified as follows.

A general dynamic system $S_{GD}$ is a quintuple Failed to parse (unknown function "\grp"): {\displaystyle ([I,O], [\lambda: I \to O], \R_S , [\Delta: \R_S \to \R_S], \grp_B)} , where:

1. $I$ and $O$ are, respectively, the input and output [[../NoncommutativeGeometry4/|manifolds]] of the system , $S_{GD}$;
2. ${ℝ}_S$ is a category with structure determined by the components of $S_{GD}$ as [[../TrivialGroupoid/|objects]] and

with the links or relations between such components as [[../TrivialGroupoid/|morphisms]];

1. $\mathrm{\Delta }:{ℝ}_S\to {ℝ}_S$ is the `dynamic transition' functor in the [[../TrivialGroupoid/|functor category]] $Au{t}_S$

of system endomorphisms (which is endowed with a [[../QuantumOperatorAlgebra5/|groupoid]] structure only in the case of reversible, [[../ThermodynamicLaws/|closed systems]]);

1. $\lambda$ is the [[../StableAutomaton/|output `function]] or map' represented as a manifold [[../TrivialGroupoid/|homeomorphism]];
2. Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} is a topological groupoid specifying the boundary, or boundaries, of $S_{GD}$.

Remark . We can proceed to define automata and certain simpler quantum systems as particular, or specialized, cases of the above general dynamic system quintuple.